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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games
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by Luca Codenotti, Marta Lewicka and Juan Manfredi PDF
Trans. Amer. Math. Soc. 369 (2017), 7387-7403 Request permission

Abstract:

We study the double-obstacle problem for the $p$-Laplace operator, $p\in [2, \infty )$. We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-of-war games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token’s position, converges to $0$. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.
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Additional Information
  • Luca Codenotti
  • Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
  • Email: luc23@pitt.edu
  • Marta Lewicka
  • Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
  • MR Author ID: 619488
  • Email: lewicka@pitt.edu
  • Juan Manfredi
  • Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
  • MR Author ID: 205679
  • Email: manfredi@pitt.edu
  • Received by editor(s): January 20, 2016
  • Received by editor(s) in revised form: April 17, 2016, and April 22, 2016
  • Published electronically: May 11, 2017
  • Additional Notes: The first and second authors were partially supported by NSF award DMS-1406730
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 7387-7403
  • MSC (2010): Primary 35J92
  • DOI: https://doi.org/10.1090/tran/6962
  • MathSciNet review: 3683112